Is it permissible for a wife to use birth control without her husband's knowledge and desire for children, and does this necessitate his forgiveness, or can the husband refrain from doing so?

A wife is not permitted to use contraception without her husband's knowledge, because procreation is a shared right between the spouses. The husband must forgive her and overlook her action, as forgiveness is among the manners of the Prophet, peace and blessings be upon him. He should also explain to her the ruling on her action so that she does not repeat it.